19. The specific gravity of an alloy made of two metals in equal volumes is 4. But when
the alloy is made up of the same two metals in equal masses, its specific gravity is 3.
The specific gravity of each metal is? The specific gravity is defined as the density of the
material compared to some reference density.
a. 8, 4
b. 6, 4
c. 6, 2
d. 4, 2
Solution Derivation – Specific Gravity Alloy Problem
Let the specific gravities (densities) of the two metals be d₁ and d₂.
Case 1: Equal Volumes
Let each volume be V.
Total mass = V(d₁ + d₂), total volume = 2V.
Hence, specific gravity of alloy = (d₁ + d₂) / 2 = 4
Therefore, d₁ + d₂ = 8.
Case 2: Equal Masses
Let each mass be m.
Volumes = m/d₁ and m/d₂.
Total mass = 2m, total volume = m(1/d₁ + 1/d₂).
Hence, specific gravity of alloy =
2m / [m(1/d₁ + 1/d₂)] = 2 / (1/d₁ + 1/d₂) = 3
So, 1/d₁ + 1/d₂ = 2/3
Rewriting, (d₁ + d₂) / (d₁ d₂) = 2/3
Substitute d₁ + d₂ = 8:
8 / (d₁ d₂) = 2/3 \
→ d₁ d₂ = 12
Forming the Quadratic Equation
x² − 8x + 12 = 0
Factoring: (x − 6)(x − 2) = 0
Hence, d₁ = 6 and d₂ = 2.
Option Analysis
| Option | Equal Volumes (Avg) | Equal Masses (Harmonic Mean) | Result |
|---|---|---|---|
| a. 8, 4 | (8+4)/2 = 6 ≠ 4 | 2×8×4 / (8+4) ≈ 5.33 ≠ 3 | Incorrect |
| b. 6, 4 | (6+4)/2 = 5 ≠ 4 | 2×6×4 / (6+4) ≈ 4.8 ≠ 3 | Incorrect |
| c. 6, 2 | (6+2)/2 = 4 ✅ | 2×6×2 / (6+2) = 3 ✅ | Correct |
| d. 4, 2 | (4+2)/2 = 3 ≠ 4 | 2×4×2 / (4+2) ≈ 2.67 ≠ 3 | Incorrect |
Conclusion
When the alloy shows specific gravity 4 for equal volumes and 3 for equal masses, the two metals have specific gravities:
d₁ = 6 and d₂ = 2.
Concept: This question tests the use of arithmetic and harmonic means to relate densities in alloy problems—a common concept in CSIR NET quantitative aptitude. The quadratic approach efficiently derives densities without trial and error.
